Low-frequency broadband vibration-damping metamaterial ground shield and analysis method for band gap characteristics thereof

Abstract

This invention discloses a vibration-damping metamaterial ground barrier capable of achieving a low-frequency, wide-bandgap design. The ground barrier comprises multiple periodically arranged unit cell structures. Each unit cell structure includes an annular rubber layer, an inner steel sheet disposed within the annular rubber layer, and an outer steel sheet disposed outside the annular rubber layer. An inner steel oscillator is disposed within the inner steel sheet, and an outer steel oscillator is disposed within the outer steel sheet. This invention also discloses an analytical method for the bandgap characteristics of the vibration-damping metamaterial ground barrier capable of achieving a low-frequency, wide-bandgap design. This invention is specifically designed for suppressing elastic waves below 20Hz, possessing not only the characteristics of a low-frequency bandgap and a large bandgap width but also structural stability. The finite element method is used to calculate the band structure and frequency response function of the ground barrier.

Description

Technical Field

[0001] This invention relates to the field of ground barrier technology, and in particular to a vibration-damping metamaterial ground barrier capable of achieving low-frequency wide bandgap and an analytical method for its bandgap characteristics. Background Technology

[0002] Currently, vibration and noise problems are widespread in engineering fields such as high-speed rail, aircraft, ships, and vehicles. Excessive vibration and noise can seriously affect the reliability, safety, and service life of equipment, and may even endanger human health and safety. In particular, low-frequency elastic waves can have a significant impact on precision instruments in high-speed rail lines. Vibration and noise propagate in structures in the form of elastic waves. In engineering, many structures exhibit a wave-like response under dynamic loads. Suppressing the transmission of low-frequency elastic waves has always been a major challenge in engineering. Therefore, the core of vibration and noise control is the regulation of elastic wave propagation in structures. Currently, common seismic protection measures for buildings include installing seismic isolation bearings and applying dampers. [1-2] These technologies are applied in the early stages of building construction, making it difficult to protect existing building structures. Furthermore, the current focus and challenge of structural vibration and noise reduction lies in the low-frequency range. Traditional passive control methods such as damping are not very effective at suppressing low frequencies, and while active control technologies primarily target low frequencies, they suffer from high costs, complex control systems, and low reliability. [3-5] .

[0003] In recent years, many scholars have used the bandgap characteristics of phononic crystals to control vibration and noise. Phononic crystals are artificially designed periodic dielectric materials, known as periodic structures. [6-10] Existing research has found that periodic structures possess unique mechanical or acoustic properties. Specifically, within certain frequency bands, elastic waves in periodic waveguides can be completely blocked from propagation without external damping or energy suppression. The frequency band corresponding to this special phenomenon is called a "bandgap." [11-15] Therefore, based on the principle of this bandgap, many scholars have designed and adjusted the shape and materials of periodic structures to achieve directional control of elastic waves, thus providing novel ideas and theoretical support for blocking vibration and noise. Currently, research on controlling rail transit vibration and noise using the bandgap characteristics of periodic structures has achieved some results, but overall it is still in its initial stage.

[0004] Domestic and international scholars have conducted extensive research on the suppression of elastic waves using periodic structures. By designing different geometric configurations and selecting different material parameters, elastic waves in different frequency ranges can be attenuated and isolated.

[16] To study Rayleigh waves, field tests were conducted: using periodic drilling in the ground, it was shown that the designed periodic structure could isolate Rayleigh waves of approximately 50 Hz in a specific direction, demonstrating the feasibility of periodic structures in practical seismic applications. Palermo et al.

[17] A single-unit resonator embedded in the soil was designed to attenuate Rayleigh seismic waves. The fundamental frequency of the dynamic characteristics of the concrete frame building was considered, and a more compact, small-unit simple resonator was used to protect the building from Rayleigh waves. (Jiankun Huang et al.)

[18] A pile barrier composed of periodically arranged hollow piles filled with soft or hard filler material was designed. This pile barrier can generate low-frequency or mid-frequency attenuation regions. (Pu et al.)

[19] A filled groove was designed, and field experiments were conducted on ground vibrations caused by trains to explore the attenuation region of surface waves in the periodic filled groove. However, the bandgap initiation frequency and bandwidth of the periodic structure designed to suppress elastic waves below 20Hz were not low enough, resulting in an unsatisfactory vibration isolation effect. [20-21] .

[0005] Moreover, most existing sound barriers are designed with relatively simple geometric structures and do not make good use of acoustic metamaterials to innovate the model. Even if the optimization design technology for sound barriers produces low-frequency band gaps, the bandwidth is very narrow. They can only be designed for specific needs, and there is no universal and efficient sound barrier that can be adjusted for different needs. Therefore, targeted vibration reduction to achieve the target has become a very urgent need.

[0006] [1] Yang Weiguo, Li Hao, Kang Kai, et al. Analysis of the whole process of structural vibration caused by underground high-speed railway and research on vibration reduction performance of vibration isolation bearings [J]. Vibration and Shock, 2022, 41(01):254-264.

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[0008] [3] Zhao Hao-jiang et al. Numerical study and topology optimization of vibration isolation support structures[J]. International Journal of Mechanical Sciences, 2022, 228.

[0009] [4] Gu Jinchao. Application of vibration reduction and noise reduction technology in urban rail transit [J]. Shanxi Architecture, 2023, 49(04):120-123.DOI:10.13719 / j.cnki.1009-6825.2023.04.031.

[0010] [5] Liu Jinhui, Liu Penghui, Yang Yiqian, et al. Test and analysis of vibration reduction and noise reduction effect of elevated track [J]. Vibration and Shock, 2022, 41(15):83-89. DOI:10.13465 / j.cnki.jvs.2022.15.011.

[0011] [6] LEFEBVRE G, ANTONAKAKIS T, ACHAOUI Y, et al. Unveiling extremeanisotropy in elastic structured media[J]. Physical Review Letters, 2017, 118(25): 254302.

[0012] [7] Qin Kaiqiang, Liu Ze, Yu Guilan. Partially embedded surface wave barrier with low-frequency ultrawide attenuation domain [J]. Earthquake Engineering and Engineering Vibration, 2020, 40(2):187-194.

[0013] [8] Wagner PR, Dertimanis VK, Chatzi EN, et al. Robust-to-uncertainties optimal design of seismic metamaterials [J]. Journal of Engineering Mechanics, 2017, 144(3): 04017181.

[0014] [9] DU QJ, ZENG Y, HUANG GL, et al. Elastic metamaterial-based seismic shield for both Lamb and surface waves[J]. AIP Advances, 2017,7(7):075015.

[0015]

[10] Pu X, Shi Z F. A novel method for identifying surface waves inperiodic structures [J]. Soil Dynamics and Earthquake Engineering, 2017, 98:67―71.

[0016]

[11] Das Sachchidanand et al. Investigations on the band-gapcharacteristics of one-dimensional flexural periodic structures with varying geometries[J]. Journal of Vibration and Control, 2022, 28(23-24): 3469-3481.

[0017]

[12] SUN FF, XIAO L. Bandgap characteristics and seismicapplications of inerter-in-lattice metamaterials[J]. Journal of Engineering Mechanics, 2019, 145(9): 04019067.

[0018]

[13] Li Lixia, Yang Jibo, Lü Ruixiang, et al. Study on low-frequency bandgap mechanism of novel annular resonant radial elastic metamaterial structure [J]. Journal of Xi'an Jiaotong University, 2020, 54(11):91-97.

[0019]

[14] He Feiyang et al. Flexural wave bandgap properties of phononiccrystal beams with interval parameters[J]. Applied Mathematics and Mechanics,2023, 44(2) : 173-188.

[0020]

[15] Al Ba'ba'a Hasan B. et al. Theory of Truncation Resonances inContinuum Rod‐Based Phononic Crystals with Generally Asymmetric Unit Cells(Adv. Theory Simul. 2 / 2023)[J]. Advanced Theory and Simulations, 2023, 6(2).

[0021]

[16] Brûlé S, Javelaud E H, Enoch S, et al. Experiments on seismicmetamaterials: molding surface waves [J].Physical Review Letters, 2014, 112(13): 133901.

[0022]

[17] Palermo A, Vitali M, Marzani A. Metabarriers with multi-masslocally resonating units for broad band Rayleigh waves attenuation [J]. SoilDynamics and Earthquake Engineering, 2018, 113: 265-277.

[0023]

[18] Jiankun Huang, Zhifei Shi. Vibration Reduction of Plane WavesUsing Periodic In-Filled Pile Barriers[J]. Journal of Geotechnical andGeoenvironmental Engineering, 2015, 141(6): 04015018-04015018.

[0024]

[19] Xingbo Pu, Zhifei Shi, Hongjun Xiang. Feasibility of ambientvibration screening by periodic geofoam-filled trenches[J]. Soil Dynamics and Earthquake Engineering, 2018, 104: 228-235.

[0025]

[20] Miniaci M, Krushynska A, Bosia F, et al. Large scale mechanicalmetamaterials as seismic shields [J]. New Journal of Physics, 2016, 18(8):083041.

[0026]

[21] Wagner PR, Dertimanis VK, Chatzi EN, et al. Robust-to-uncertainties optimal design of seismic metamaterials [J]. Journal of Engineering Mechanics, 2017, 144(3): 04017181. Summary of the Invention

[0027] To address the problems existing in the prior art, the purpose of this invention is to provide a vibration-damping metamaterial ground barrier that can achieve low-frequency wide bandgap and an analytical method for its bandgap characteristics. This invention is specifically designed to suppress elastic waves below 20Hz, and not only has the characteristics of low-frequency bandgap and large bandgap width, but also structural stability. Furthermore, the finite element method is used to calculate the energy band structure and frequency response function of the ground barrier.

[0028] To achieve the above objectives, the technical solution adopted by the present invention is: a vibration-damping metamaterial ground barrier capable of achieving low-frequency wide bandwidth, the ground barrier comprising multiple periodically arranged unit cell structures, each unit cell structure comprising an annular rubber and an inner steel sheet disposed inside the annular rubber and an outer steel sheet disposed outside the annular rubber, the inner steel sheet containing an inner steel vibrator and the outer steel sheet containing an outer steel vibrator.

[0029] As a further improvement of the present invention, the inner steel sheet is provided with a pair of inner steel vibrators, which are symmetrically arranged at both ends of the inner steel sheet, and the pair of inner steel vibrators and the inner steel sheet form an "8" shape structure; the outer steel sheet is provided with four outer steel vibrators, which are respectively arranged in the middle of the four sides of the outer steel sheet, and the four outer steel vibrators extend from the side surface of the outer steel sheet to contact the outer wall of the annular rubber.

[0030] As a further improvement of the present invention, both the internal steel vibrator and the external steel vibrator are hollow structures, and the hollow structures are filled with particulate damping material.

[0031] As a further improvement of the present invention, the particulate damping material in the cavity structure has a filling rate of 80%, a porosity of 0.367, and an apparent density of 3975 kg / m³. 3 .

[0032] This invention also provides an analytical method for the bandgap characteristics of a vibration-damping metamaterial ground barrier that can achieve a low-frequency wide bandgap, as described above, comprising the following steps:

[0033] Step 1: Using the finite element method, and based on the Bloch-Floquet theorem, apply Bloch boundary conditions (i.e., displacement magnitude conditions) in the X and Y directions of the unit cell structure:

[0034]

[0035] In the formula: i is an imaginary number, r is a displacement vector; a is a periodic constant; k is the wave vector of the reciprocal lattice space.

[0036] In finite element calculations, by introducing the Bloch wave vector k value, a set of structural eigenfrequency can be obtained; by scanning the wave vector k along the irreducible Brillouin zone, the natural frequency of the unit cell structure in this direction can be obtained; by arranging the natural frequencies in different directions according to their directions, the energy band diagram of the earth barrier structure can be obtained, and the vibration modes at the starting frequency point A and the cutoff frequency point B in the energy band diagram can be extracted.

[0037] Step 2: Simplify the resonance mode of the initial frequency into a mass spring system, calculate the shear stiffness of the annular rubber using formulas from mechanics of materials, and calculate the initial and cutoff frequencies using different shear stiffnesses.

[0038] As a further improvement of the present invention, step 2 is specifically as follows:

[0039] Let k1 be the outer ring shear stiffness, k2 be the inner ring shear stiffness, m1 be the mass of the outer steel plate, and m2 be the sum of the masses of the inner steel plate and the annular rubber; the calculation is as follows:

[0040]

[0041]

[0042] Where e, c, f, d, and H are all geometric parameters of the new structure, and E and These are Young's modulus and Poisson's ratio of the rubber material, respectively.

[0043] The mass of the outer steel sheet is:

[0044]

[0045] The quality of the internal steel sheet and the ring rubber and for:

[0046]

[0047] The equations of motion for the simplified model based on the initial frequency are:

[0048]

[0049] The equations of motion for the simplified model with cutoff frequency are:

[0050]

[0051]

[0052] In the formula, represents the displacement of the inner steel-rubber composite layer; therefore, the starting and cutting frequencies of the fully gapped ground barrier are shown in the following formula:

[0053]

[0054] .

[0055] As a further improvement to the present invention, the following steps are also included:

[0056] Step 3: Keeping other parameters constant, adjust the geometric parameters of the ground barrier to analyze its band gap characteristics.

[0057] The beneficial effects of this invention are:

[0058] This invention, based on the concepts of periodic structures and mechanical metamaterials, designs an embedded figure-eight discrete vibrating ring ground barrier for low-frequency (below 20Hz) elastic waves, innovating upon the traditional geometric structure of ground barriers. Using the finite element method and Bloch's principle, numerical simulations were performed on the barrier's band structure and frequency response curves. The influence of variable frequency damping on the structural bandgap was analyzed. The results show that the designed ground barrier effectively attenuates and isolates low-frequency elastic waves below 20Hz, overcoming the shortcomings of current sound barriers that cannot generate low-frequency bandgap and have excessively narrow bandgap. Attached Figure Description

[0059] Figure 1 This is a model diagram of an embedded figure-eight discrete vibration ring ground barrier in an embodiment of the present invention;

[0060] Figure 2 This is a side view of a single-cell structure in an embodiment of the present invention;

[0061] Figure 3 This is a front view of the unit cell structure in an embodiment of the present invention;

[0062] Figure 4 This is a schematic diagram of the particle damping structure in an embodiment of the present invention;

[0063] Figure 5 This is a graph of the frequency conversion damping function in an embodiment of the present invention;

[0064] Figure 6 This is a Bloch wave vector k-scan region diagram in an embodiment of the present invention;

[0065] Figure 7 This is a bandgap characteristic diagram from an embodiment of the present invention;

[0066] Figure 8 These are vibration mode diagrams at starting frequency point A and ending frequency point B in an embodiment of the present invention.

[0067] Figure 9 This is a schematic diagram illustrating the influence of the width of the internal steel oscillator in an embodiment of the present invention;

[0068] Figure 10 This is a schematic diagram illustrating the influence of the width of the annular rubber ring in an embodiment of the present invention;

[0069] Figure 11 This is a comparison diagram of the dispersion curve and the frequency response function curve in an embodiment of the present invention;

[0070] Figure 12 This is a comparison diagram of the ground barrier vibration modes under different operating conditions at 20Hz in an embodiment of the present invention;

[0071] Figure 13 The diagram shows four frequency mode shapes within the bandgap in an embodiment of the present invention. Detailed Implementation

[0072] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0073] Example

[0074] The design of the embedded figure-eight discrete vibration ring ground barrier consists of five core steps.

[0075] The first step is to design a wave barrier capable of generating a low-frequency bandgap. This involves combining various geometries of ground barriers to design, such as... Figures 1-3 The optimal structure is shown.

[0076] The unit cell consists of an inner steel oscillator, a ring-shaped rubber element, an outer steel oscillator, and a ring-shaped steel sheet, arranged periodically to form a ground barrier. Based on the original ground barrier, the original outer ring and inner oscillator are hollowed out, replaced with a cavity structure, and then filled with... Figure 4 The granular damping material is randomly filled with particles to form a granular damped oscillator ring assembly. Considering that the cavity structure of the phonon crystal ground barrier is in a vibrating state during operation, and to maintain a certain amount of porosity while maximizing the optimal performance of granular damping, the particle filling rate is selected as 80%, the porosity is set to 0.367, and the apparent density of the granular damping filler is designed to be 7850×(1-0.367)×0.8=3975 kg / m3. The function curve of the variable frequency granular damping is shown below. Figure 5 As shown, while keeping the original mass of the structure unchanged, the particles achieve the effect of dissipating energy during the vibration of the earth barrier.

[0077] The geometric parameters of the unit cell are as follows: inner steel sheet a=0.5m, b=0.8m; side length of inner square steel oscillator a0=0.25m; annular rubber c=1m, d=1.4m; outer steel sheet e=1.5m, f=2m; outer steel oscillator b0=0.4m; spacing between inner steel oscillators d1=0.125m, d2=0.1m.

[0078] Step two: According to the Bloch-Floquet theorem, apply Bloch boundary conditions, i.e., displacement magnitude conditions, in the X and Y directions of the element structure:

[0079]

[0080] In the formula: r is the displacement vector; a is the period constant; k is the wave vector of the reciprocal lattice space.

[0081] In finite element method (FEM) calculations, the Bloch wave vector k value is introduced to obtain a set of structural eigenfrequencys. Since the element is a centrosymmetric structure, the wave vector k can be obtained simply by scanning along the irreducible Brillouin zone ("M→Γ→X→M"). Figure 6As shown, the natural frequencies of the structure in this direction can be solved by following the order from to . Arranging the natural frequencies in different directions according to their directions yields the following results. Figure 7 The energy band diagram of the barrier structure is shown. To investigate the mechanism of band gap formation, the vibrational modes at the starting frequency point A and the cutoff frequency point B in the energy band structure diagram are extracted, as follows: Figure 8 As shown;

[0082] Step three involves further investigating the boundary frequencies of the bandgap in the ground barrier vibration bandgap by continuing the bandgap characteristic analysis of the cell. As shown in the guided wave mode shapes above, the guided wave mode determining the initial frequency exhibits an anti-phase resonance between the outer steel sheet and the rubber, while the vibration of the inner steel sheet is almost zero. Therefore, the initial frequency fs is approximately equal to the resonance frequency of the outer steel sheet. This resonance mode can be simplified as a mass-spring system. The outer steel sheet is considered as a mass block m1, and a linear spring k1 is used to simulate the shear stiffness of the rubber, thus establishing a simplified model. The guided wave mode determining the cutoff frequency exhibits an anti-phase resonance between the connection between the inner steel sheet and the rubber and the outer steel sheet, with relatively small vibration of the outer steel sheet, and a cutoff frequency fs. e This is approximately equal to the anti-phase resonant frequency. This resonance mode can also be simplified to a mass-spring system. The outer steel sheet is considered as mass m1, and the inner steel sheet as mass m2. A linear spring k1 is used to simulate the shear stiffness of the rubber, thus establishing a simplified model. As shown above, this bandgap is a locally resonant bandgap. Using formulas from mechanics of materials, the shear stiffness of the rubber can be calculated. To eliminate errors, different shear stiffnesses are used to calculate the initial and cutoff frequencies:

[0083]

[0084]

[0085] Quality of outer steel sheet for:

[0086]

[0087] The quality of the inner steel sheet and rubber and for:

[0088]

[0089] The equations of motion for the simplified model based on the initial frequency are:

[0090]

[0091] In the formula, u1 represents the displacement of the outer steel sheet. The equation of motion for the simplified model at the cutoff frequency is:

[0092]

[0093]

[0094] In the formula, u2 represents the displacement of the inner steel-rubber composite layer. Therefore, the start and cutoff frequencies of the fully gapped ground barrier are shown in the following formula:

[0095]

[0096]

[0097] Substituting the parameters from the first section, the calculation result is: Comparing the calculation results using the simplified model with those using the finite element model, the starting frequency is the same, and the error in the cutoff frequency is only 3Hz. Therefore, the simplified model can be used to calculate the starting and cutoff frequencies of the full bandgap.

[0098] Step four: To investigate the impact of changes in the geobar's geometric parameters on the band gap, while keeping other parameters constant, the geometric parameters of the geobar were adjusted: the length and width of the inner steel plate, outer steel plate, and annular rubber, and the side length of the inner square steel oscillator. These adjustments correspondingly altered the geobar's band gap, achieving the desired effect according to the needs of different regions. Five geobars with different parameters were selected for comparison of band gap changes. Figure 9 The graph shows the variation of the bandgap initiation frequency, cutoff frequency, and bandgap width when the width of the internal steel oscillator is a = 0.3m, 0.5m, 0.7m, 0.8m, and 0.9m. Figure 10 The graph shows the variation of the bandgap initiation frequency, cutoff frequency, and bandgap width when the width of the annular rubber ring is c = 0.6m, 0.8m, 1m, 1.2m, and 1.3m. It can be concluded that as the width of the internal steel oscillator increases, the overall stiffness of the structure increases, leading to an increase in frequency. Increasing the width of the annular rubber ring leads to a decrease in stiffness and an increase in flexibility; changes in both mass and stiffness result in a decrease in the bandgap frequency. To verify the bandgap generated by the figure-eight discrete ring ground barrier, a structure of six units arranged periodically along the X-direction was used. A boundary load of 1 N / m² was applied in the X-direction to excite body waves and elastic waves propagating in the soil. To prevent the influence of displacement below the soil, a fixed constraint was applied below the soil. The displacement amplitude response was picked up at the right side of the soil, and the frequency response function curve is shown below. Figure 11 As shown in the figure, the displacement response attenuated within the frequency range of 5~22Hz, with a total attenuation range of up to 17Hz. The results are basically consistent with the Γ-M direction dispersion curve shown in the figure, verifying the accuracy of the bandgap calculation for the earth barrier structure.

[0099] Step five concludes by further comparing the vibration attenuation effects of applying damping to the embedded figure-eight discrete vibration ring ground barrier. Figure 12 The displacement fields are represented at 20Hz for an undamped ground barrier, a damped ground barrier, and a soil-only field. Figure 12(a) shows the transmission of elastic waves in the soil without a ground barrier, with obvious transmission of elastic waves on the soil surface and inside. Figure 12 (b) in the diagram represents an undamped figure-eight discrete vibration ring ground barrier, and... Figure 12 The comparison in (a) shows that the elastic wave was almost absorbed by the earth barrier, and there was almost no vibration on the right side. Figure 12 (c) in the diagram represents a damped figure-eight shaped discrete vibrating ring ground barrier, and... Figure 12 Compared to (b) in the middle, the elastic wave is almost completely absorbed, with a significant effect. Figure 12 This indicates that the ground barrier can effectively prevent the propagation of elastic waves within the bandgap frequency range, and the effect is even better and the seismic isolation effect is more significant if damping is applied.

[0100] To further illustrate the effects of this invention, Figure 13 The middle section shows the vibration modes of the structure at different frequencies within the bandgap. Figure 13 Figures (a), (b), (c), and (d) show the mode shapes at 5Hz, 10Hz, 15Hz, and 20Hz, respectively. At 5Hz, the vibration amplitude reaches its maximum at the location where the boundary load is applied, and then decreases significantly with the incident direction. Low transmittance is observed on the right side, and the cell group on the right side hardly vibrates. At 10Hz, the vibration of the structure reaches its maximum at the upper left corner, and the cells hardly vibrate. The vibration mode is antisymmetric along the incident direction, consistent with the results in the figure. At 15Hz, the vibration is mainly concentrated in the upper right corner, which is consistent with the antisymmetric mode of wave vibration. At 20Hz, the vibration is symmetric on the incident surface, the vibration of the cell group is almost zero, and the exit surface part vibrates.

[0101] It can be concluded that the fully embedded ground barrier, formed by periodically arranged elements within the soil, exhibits high strength and allows for flexible adjustment of its parameters to alter the bandgap range under varying environmental conditions. With damping, the barrier can essentially cover elastic waves below 20Hz. It demonstrates excellent attenuation for low-frequency elastic waves, with a total attenuation range reaching 17Hz. The bandgap opening is attributed to the local resonance of the periodic structure. Damping and filling the figure-eight discrete ring structure allows for adjustment of the bandgap range and the attenuation effect of elastic waves within the bandgap, creating an ultra-low frequency wide bandgap. The figure-eight discrete ring periodic elastic wave barrier effectively attenuates the acceleration response of elastic waves, providing a new design approach for the application of ground barriers in practical engineering (especially high-speed railway engineering).

[0102] The embodiments described above are merely illustrative of specific implementations of the present invention, and while the descriptions are detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. An analysis method of a bandgap characteristic of a vibration-damping metamaterial ground screen capable of realizing a low-frequency broadband gap, characterized by, The ground barrier comprises multiple periodically arranged unit cell structures. Each unit cell structure includes an annular rubber ring, an inner steel sheet disposed inside the annular rubber ring, and an outer steel sheet disposed outside the annular rubber ring. An inner steel vibrator is disposed within the inner steel sheet, and an outer steel vibrator is disposed within the outer steel sheet. The method includes the following steps: Step 1: Using the finite element method, and based on the Bloch-Floquet theorem, apply Bloch boundary conditions (i.e., displacement magnitude conditions) in the X and Y directions of the unit cell structure: ； In the formula: i is an imaginary number, r is a displacement vector; a is a periodic constant; k is the wave vector of the reciprocal lattice space; In finite element calculations, by introducing the Bloch wave vector k value, a set of structural eigenfrequency can be obtained; by scanning the wave vector k along the irreducible Brillouin zone, the natural frequency of the unit cell structure in this direction can be obtained; by arranging the natural frequencies in different directions according to their directions, the energy band diagram of the earth barrier structure can be obtained, and the vibration modes at the starting frequency point A and the cutoff frequency point B in the energy band diagram can be extracted. Step 2: Simplify the resonance mode of the initial frequency into a mass spring system, use the formulas of mechanics of materials to calculate the shear stiffness of the annular rubber, and use different shear stiffnesses to calculate the initial frequency and cutoff frequency. Step 2 is described in detail below: Let k1 be the outer ring shear stiffness, k2 be the inner ring shear stiffness, m1 be the mass of the outer steel sheet, and m2 be the sum of the masses of the inner steel sheet and the annular rubber ring; the calculation is as follows: ； ； Where e, c, f, d, and H are all geometric parameters of the new structure, and E and These are Young's modulus and Poisson's ratio of the rubber material, respectively. The mass of the outer steel sheet is: ； the mass of the inner steel sheet and the annular rubber and is: ； The equations of motion for the simplified model based on the initial frequency are: ； The equations of motion for the simplified model with cutoff frequency are: ； ； In the formula, u2 represents the displacement of the inner steel-rubber composite layer; therefore, the start and cutoff frequencies of the fully gapped ground barrier are shown in the following formula: ； ； It also includes the following steps: Step 3: Keeping other parameters constant, adjust the geometric parameters of the ground barrier to analyze its band gap characteristics.

2. The method for analyzing the bandgap characteristics of a vibration-damping metamaterial ground barrier capable of achieving a low-frequency, wide-bandgap structure according to claim 1, characterized in that, The inner steel sheet contains a pair of inner steel vibrators, which are symmetrically arranged at both ends of the inner steel sheet, forming an "8" shape with the inner steel sheet. The outer steel sheet contains four outer steel vibrators, which are respectively arranged in the middle of the four sides of the outer steel sheet, and extend from the side surface of the outer steel sheet to contact the outer wall of the annular rubber.

3. The method for analyzing the bandgap characteristics of a vibration-damping metamaterial ground barrier capable of achieving a low-frequency, wide-bandgap structure according to claim 1, characterized in that, Both the internal and external steel vibrators are hollow structures, and the hollow structures are filled with particulate damping material.

4. The method for analyzing the bandgap characteristics of a vibration-damping metamaterial ground barrier capable of achieving a low-frequency, wide-bandgap structure according to claim 3, characterized in that, The filling rate of the particle damping material in the cavity structure is 70-90%, the porosity is 0.315-0.36, and the apparent density is 3478.13-4471.88 kg / m 3 .

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